Sudoku is a mathematic puzzle that has gained considerable popularity in the 21st century, rivalling the classic puzzle that is the crossword. You are given a 9×9 table divided into 9 equal squares, filled with a certain number of digits. Your goal is to fill in the table so that each row, column and subsquare (of 9 small squares) contains every digit from 1 to 9. You are not allowed to have the same number appear on the same row, column or subsquare, as there are not enough spaces for spare digits.
The more digits (“clues”) that you are given at the start of the puzzle, the easier it is to solve it. This begs the question: what is the minimum number of clues that you need to solve a sudoku puzzle?
Sudoku puzzles with 17 clues have been completed traditionally. We know that 7 clues is not enough as the last 2 digits can be interchanged, creating puzzles with more than one solution. Using mathematics, we know that if we can solve a puzzle with n clues, then a puzzle with n+1 clues can be solved as well. Ergo, the answer lies somewhere between 8 and 16.
In 2012, Gary McGuire, Bastian Tugemann and Gilles Civario tackled this problem using one of the oldest tricks in mathematical analysis: brute force. The total number of possible sudoku puzzles that can be generated is 6,670,903,752,021,072,936,960, or 6.67 x 10²¹. After accounting for symmetry arguments (meaning that two puzzles may be essentially identical, but just rotated or flipped), we are left with 5,472,730,538 possible unique solutions.
The team used supercomputers to analyse all of these possibilities to see if any puzzle can be solved with just 16 clues, as the conventional thought was that 17 was the minimum number of clues possible from traditional methods. After a year of calculations, the computer found no sudoku puzzle could be solved with only 16 clues. This was confirmed by another team from Taiwan a year later, proving that the minimum number of clues required for sudoku is indeed 17.
Alan Turing was a brilliant British mathematician who was pivotal in cracking the German Enigma cipher using a complex computing machine. He was highly influential in the founding of computing science. One of his greatest areas of interest was artificial intelligence. Like other computer scientists of the time, Turing predicted that machine intelligence was possible in the future with rapid development of computers. On this topic, he proposed the following question: at what point is a machine truly “intelligent”?
Intelligence is too complicated to define neatly in a single line. Therefore, here is a simpler question: can a machine do what we can do? For this, he proposed a thought experiment based on a party game known as the imitation game. In the imitation game, a man and a woman go into separate rooms. Guests then try to tell who is a man and who is a woman by writing a series of questions, slipping it under the door, then receiving a typewritten answer. If the guests cannot tell the two apart, the two win the game.
Turing modified this game into what is now known as the Turing test. He proposed replacing one person with a machine. A person and a computer are placed in separate rooms and are asked the same question by a judge. They then give a typed response. If the judge cannot confidently tell who is human and who is not, then the machine passes the Turing test.
Of course, the Turing test was not designed as a formal assessment and is merely a thought experiment. It has plenty of weaknesses, such as the fact that it only tests whether the machine is acting “like a human” rather than “intelligently”. For example, some computers have passed the Turing test by intentionally making typos to mimic human behaviour. Some have argued that machines that pass the Turing test do not truly exhibit intelligence, as it is impossible to tell if they fully understand the language or whether they are just running algorithms on symbols that the machine does not understand.
Regardless, the Turing test opens the door on the exciting yet frightening world of artificial intelligence and what the future holds for humanity.
Author Bernard Werber (the inspiration for this Encyclopaedia) posited the following theory: if we could see the future, would we not actively build towards a better future? Imagine a tree soaring high into the sky, stretching countless branches in all directions. The many branches of the tree branch off into smaller branches, which branch into even more smaller branches. At the end of each branch, there hangs a leaf. This tree is not a normal tree; it is a Tree of Possibilities that represents the flow of time from the beginning of the universe to the distant future. Each split in a branch represents the creation of two different futures due to a choice or a change, while a leaf represents the final future created from the cumulative effects of these changes. Thus, the Tree of Possibilities is the ultimate crystal ball showing all the pasts that could have been and all the futures that can happen.
Of course the Tree of Possibilities is a fictional model created in our imaginations. But what if we could actually make this tree? First, we would create an organisation of the greatest scientists, mathematicians, sociologists, psychologists, historians, philosophers, science fiction writers etcetera that represent the many fields of knowledge. These people are gathered in a location far from the reaches of governments and the media, where they can discuss without any interference. These specialists will debate over all sorts of topics, amalgamating their knowledge and intuition to generate a tree diagram as mentioned above. This is a diagram free from ethics, morals, laws, optimism, pessimism and individualism – the ultimate objective view of all possible futures that humanity and the Earth may face. The experts may agree with each other at times and disagree at times. There is ample possibility that their postulations are wrong. But none of these matter. The important point is not that the Tree is “accurate” or not, but that it is an extensive scenario database of all the paths humanity can walk on towards the future.
The Tree of Possibilities will have various conjectures such as: What if nuclear war broke out? What if artificial intelligence is perfected? What if chimpanzees reach the intelligence levels of human beings? What if we build cities on the Moon? However, the future is altered much more easily that you would think. Thus, there will also be branches representing much more trivial and ordinary (even bizarre) postulations as well: What if smoking is banned? What if the average age women gave birth is older? What if rhinoceroses were domesticated pets? What if pianos do not exist?
On analysing these numerous postulations, a branch bearing the leaf with the ideal future will be found. Ergo, we can choose to follow a path of least resistance, where all the choices we make will ultimately lead to that ideal future. Essentially, the Tree of Possibilities is a tool that is used to predict the future. However, it is not “fortune telling” as it is based on logic rather than magic and divinity to see into the future. The future the Tree tells is not a set “destiny”, but rather one “possibility”. Thus, instead of fearing the future like we do with fortunes, we would instead feel excitement over the potential of finding the ideal future. If the path we are currently on is fated to an unhappy ending, then we can simply jump onto a different path with the guidance of the Tree. Unlike fortune telling, which destroys all uncertainty and any other possibilities in the future, the Tree of Possibilities provides humanity with the greatest gift: dreams of a better future.
As you could imagine, the possibilities of the future are infinite so a drawn-out diagram of the Tree of Possibilities would take up extensive amounts of space. Ergo, the ideal form of the Tree of Possibilities would be a computer program. As computer programs only need sufficient storage space, it provides a perfect environment in which the Tree may grow. The program would generate a Tree based on the information provided by the scholars, drawing out each branch and leaf, while also calculating the effects of any action on each of the possible futures. If we further applied the engine used in chess programs to predict the next few moves, then we may be able to create a program that can calculate the ideal future and the path of least resistance for humanity.
My ideal future is this. There is an isolated island, far from any interference, with a large building. At the centre of this building, there lies a supercomputer running The Tree of Possibilities. The computer is surrounded by lecture theatres, conference rooms and residential areas. Thus, specialists of each field may come to stay and use their knowledge to water the Tree and foster it. This island will provide humanity with hopes and dreams, leading them towards the best possible future based on logic and imagination.
The Tree of Possibilities will radically change our day-to-day lives. One of the greatest weaknesses of human beings is the inability to see the long-term happiness and sacrificing it for short-term gain. However, if we were able to see precisely how our actions will affect the future, then would we not act differently? Armed with insight and foresight, people will understand what is best for the future, and instead of the current near-sighted attitude of only seeing the gain right before our eyes, they will act in the best interests of their children and grandchildren. Politicians will see how useless bickering over trifling issues is and instead focus on policies that take a while to show the effects (yet nonetheless important), such as environmental conservation. The Tree of Possibilities will help us make rational decisions to create a world that the future generation will be happy living in, without being swayed by emotions and selfish greed. And so, we will build towards a utopia.
The greatest weapon a person has is imagination that can build the future.
A man asked how old a man’s three daughters were. The father replied with the following statement.
“The product of their ages is 36.”
“It’s hard to determine their ages from just that.” the man asking replied.
“The sum of their ages is same as the number of my house.”
“I still can’t figure out the answer!” the man replied again.
“My eldest daughter is blonde.” the father said, and the man, now smiling, replied.
“Oh, is that so? Then I can figure out how old your daughters are.”
How old is each daughter? And how did the man figure it out?
A computer cannot solve this problem, as it can only be solved using human logic.
The first-ever recorded computer bug occurred in 1947, on a US Navy Mark II Aiken Relay Calculator at Harvard University. After a thorough check of the computer’s software and components, operators discovered a moth that had fried itself on the circuit board, interfering with signals. The operators taped the said moth to a computer log, with the description “First actual case of bug being found”, and also added the comment that they had “debugged the computer”, coining both the terms “bug” and “debug” that are frequently used nowadays.
Although actual insects are a rare cause of problems in modern computers, the word “bug” is still the colloquial term for an error.
